Algebras of singular integral operators with kernels controlled by multiple norms
|
|
- Joan Lawrence
- 5 years ago
- Views:
Transcription
1 Algebras of singular integral operators with kernels controlled by multiple norms Alexander Nagel Conference in Harmonic Analysis in Honor of Michael Christ
2 This is a report on joint work with Fulvio Ricci, Elias M. Stein, and Stephen Wainger.
3 Calderón-Zygmund kernels A Calderón-Zygmund kernel on R n associated with the dilations λ a (x) = λ a (x 1,..., x n ) = (λ a1 x 1,..., λ an x n ) is a tempered distribution K on R n with the following properties. 1. K is given away from the origin by integration against a smooth function. 2. For every multi-index α = (α 1,..., α n ) N n and all x 0 α K(x) ( ) Cα x1 1 a x n 1 Qa n j=1 an ajαj where Q a = a a n is the homogeneous dimension. 3. K satisfies appropriate cancellation conditions so that in particular the Fourier transform K = m is a bounded function. K will often have compact support or have rapid decay outside the unit ball.
4 Homogeneous nilpotent Lie groups G denotes a homogeneous nilpotent Lie group with underlying manifold R n and automorphic dilations λ d (x 1,..., x n ) = ( λ d1 x 1,..., λ dn x n ) with d 1 d 2 d n. For x = (x 1,..., x n ), y = (y 1,..., y n ) G x y = ( x 1 + y 1, x 2 + y 2 + M 2 (x, y),..., x n + y n + M n (x, y) ) where M j (x, y) is a polynomial vanishing if x = 0 or y = 0 and M j ( λd (x), λ d (y) ) = λ dj M j (x, y). Convolution on G is given by f g(x) = f(x y 1 )g(y) dy = R n f(y)g(y 1 x) dy. R n
5 Example: The Heisenberg group The m-dimensional Heisenberg group H m has underlying space R m R m R with automorphic dilations λ 1,1,2 (x, y, t) = ( λx, λy, λ 2 t ) and group multiplication (x 1, y 1, t 1 ) (x 2, y 2, t 2 ) = ( x 1 + x 2, y 1 + y 2, t 1 + t 2 2 x 1, y x 2, y 1 ) where x, y is the Euclidean inner product. If then X j = xj + 2y j t, Y j = yj 2x j t, T = t, {X 1,..., X n, Y 1,..., Y n, T } is a basis for the left-invariant vector fields on H n.
6 The problem Consider two different dilation structures on G given by λ a (x 1,..., x n ) = (λ a1d1 x 1,..., λ andn x n ) with a 1 a 2 a n, λ b (x 1,..., x n ) = (λ b1d1 x 1,..., λ bndn x n ) with b 1 b 2 b n. Let K a and K b be Calderón-Zygmund kernels on R n with compact support associated with these dilations. For ϕ S(R n ) set T a [ϕ] = ϕ K a and T b [ϕ] = ϕ K b. What are the mapping properties of the operator T a T b on various function spaces? What can be said about size and cancellation properties of the Schwartz kernel of T a T b? Are there reasonably small algebras of convolution operators containing Calderón-Zygmund kernels associated to both dilations?
7 Background and Motivation Let Ω C n+1 be a strictly pseudoconvex domain and let = + : L 2 (0,1) (Ω) L 2 (0,1) (Ω). Given f C (0,1) (Ω), the -Neumann problem is to solve u = f on Ω u ρ = 0 on Ω u ρ = 0 on Ω. Greiner and Stein (1977) constructed a parametrix for this boundary value problem which involves the composition of two kinds of operators: operators of elliptic type coming from Poisson integrals and the Green s function for (which is essentially the Laplacian); operators of non-isotropic type coming from inverting the sub-laplacian L = n j=1 (X2 j + Yj 2 ) on the Heisenberg group H n.
8 Phong and Stein (1982) studied compositions of convolution operators on R d R T E [ϕ] = ϕ K E and T H [ϕ] = ϕ K H where is either Euclidean or Heisenberg convolution, and K E and K H are Calderón-Zygmund kernels on R d R associated with the two different dilations: λ E (x, t) = (λx, λt) and λ H (x, t) = (λx, λ 2 t). The local (near 0) and the global (near infinity) behavior of compositions T E T H or T H T E are different. Phong and Stein established: necessary and sufficient conditions for the compositions of such operators to be of weak-type (1,1); boundedness of the composition on non-isotropic Lipschitz spaces.
9 On the Heisenberg group H n, T = t and the sub-laplacian is L = n (Xj 2 + Yj 2 ). j=1 Müller, Ricci, and Stein (1995) studied general functions of L and T, and in particular established Marcinkiewicz-type theorems for multiplier operators m(l, T ) where α ξ β τ m(ξ, τ) ξ α τ β. Such multiplier operators are bounded on L p (H n ) for 1 < p <. The corresponding kernels satisfied differential inequalities α z β t K(z, t) z 2n α ( z 2 + t ) 1 β. This last estimate reflects the multi-parameter structure but is stronger than a product estimate.
10 A flag kernel associated with the dilations λ a (x) = (λ a1 x 1,..., λ an x n ) is a tempered distribution K with appropriate cancellation and singularities on an increasing sequence of subspaces: α K(x) ( ) x1 1 a a1(1+α 1 1)( ) x1 1 a 1 + x 2 1 a a2(1+α 2) 2 n = ( x 1 1 a 1 + x ) a a x j 1 j 1 + xj 1 aj(1+α j) a j. j=1 N., Ricci, Stein (2001) studied such operators when they arise in the study of the Bergman projection in tube domains over polyhedral cones and in solving b on certain quadratic submanifolds having the structure of step-2 nilpotent Lie groups. N., Ricci, Stein, Wainger (2012) and G lowacki (2010), (2013) extended this theory to homogeneous nilpotent Lie groups G of higher step, established boundedness in L p (G) for 1 < p <, and showed that such operators form an algebra under convolution.
11 A Step 3 Example Let K 1 and K 2 be Calderón-Zygmund kernels with compact support associated with the dilations λ 1 (ξ, η, τ) = (λξ, λη, λτ) and λ 2 (ξ, η, τ) = (λξ, λ 2 η, λ 3 τ). Thus α x β y γz K 1 (x, y, z) ( x + y + z ) α β γ, α x β y γz K 2 (x, y, z) ( x + y z 1 3 ) α 2β 3γ. The Fourier transforms m 1 = K 1 and m 2 = K 2 are smooth bounded functions satisfying the differential inequalities α ξ β η γ τ m 1 (ξ, η, τ) ( 1 + ξ + η + τ ) α β γ, α ξ β η γ τ m 2 (ξ, η, τ) ( 1 + ξ + η τ 1 3 ) α 2β 3γ.
12 Using Euclidean convolution, let T j [ϕ] = ϕ K j so T 1 T 2 is convolution with a distribution K and K(ξ, η, τ) = m(ξ, η, τ) = m 1 (ξ, η, τ)m 2 (ξ, η, τ). To estimate derivatives of m note that ξ gains (1 + ξ + η + τ ) 1 or (1 + ξ + η τ 1 3 ) 1, η gains (1 + ξ + η + τ ) 1 or (1 + ξ 2 + η + τ 2 3 ) 1, τ gains (1 + ξ + η + τ ) 1 or (1 + ξ 3 + η τ ) 1. Using just the product formula, the best estimates for m are ξ α η β τ γ m(ξ, η, τ) ( 1 + ξ + η 1 1 ) α 2 + τ 3 ( 2 ) β 1 + ξ + η + τ 3 ( ) γ. 1 + ξ + η + τ How should one think about such estimates?
13 One possibility is to use the theory of flag kernels and multipliers. If α ξ η β τ γ m(ξ, η, τ) ( 1 1 ) α 1 + ξ + η 2 + τ 3 ( 2 ) β 1 + ξ + η + τ 3 ( ) γ 1 + ξ + η + τ then α ξ β η γ τ m(ξ, η, τ) ξ α ( ξ + η ) β ( ξ + η + τ ) γ, ( ξ + η τ 1 3 ) α ( η + τ 2 3 ) β τ γ. Thus m is a flag multiplier for two opposite flags (0) {(ξ, 0, 0)} {(ξ, η, 0)} {(ξ, η, τ)} = R 3, (0) {(0, 0, τ)} {(0, η, τ)} {(ξ, η, τ)} = R 3.
14 The distribution K = K 1 K 2 is thus a flag kernel satisfying α x y β z γ K(x, y, z) x 1 α ( x + y ) 1 β ( x + y + z ) 1 γ, ( x + y z 1 3 ) 1 α ( y + z 2 3 ) 1 β z 1 γ. for the two opposite flags (0) {(x, 0, 0)} {(x, y, 0)} {(x, y, z)} = R 3, (0) {(0, 0, z)} {(0, y, z)} {(x, y, z)} = R 3.
15 In the 2-step case on R n R, the differential inequalities α x β t K(x, t) ( x + t ) n α ( x 2 + t ) 1 β are equivalent to the pair of flag inequalities α x β t K(x, t) x n α ( x 2 + t ) 1 β ( x + t ) n α t 1 β and However in step 3, the two-flag kernel inequalities from the last slide, α x y β z γ K(x, y, z) x 1 α ( x + y ) 1 β ( x + y + z ) 1 γ, ( x + y z 1 3 ) 1 α ( y + z 2 3 ) 1 β z 1 γ seem to give no information when x = z = 0. These flag estimates, together with the flag cancellation conditions, do imply that K is singular only at the origin.
16 A second possible way of thinking about the inequalities α ξ η β τ γ m(ξ, η, τ) ( 1 1 ) α 1 + ξ + η 2 + τ 3 ( 2 ) β 1 + ξ + η + τ 3 ( ) γ 1 + ξ + η + τ is to observe that the ξ, η, and τ derivatives are controlled by different norms and hence different families of dilations: ξ N 1 (ξ, η, τ) = ξ + η τ 1 3 η N 2 (ξ, η, τ) = ξ + η + τ 2 3 η N 3 (ξ, η, τ) = ξ + η + τ
17 The class P(E) We introduce a class P(E) of distributions on R n singular only at the origin and depending on an n n matrix E. We study: A. Properties of distributions K P(E) a. Significance of the rank of E b. Characterization using the Fourier transform c. Marked partitions d. Characterizations via dyadic decompositions e. Two-flag kernels B. Convolution operators T K ϕ = ϕ K on a homogeneous nilpotent groups a. Continuity on L p (G) b. P(E) is an algebra c. Composition of Calderón-Zygmund kernels with different homogeneities
18 Let 1 e(1, 2) e(1, 3) e(1, n) e(2, 1) 1 e(2, 3) e(2, n) E = e(3, 1) e(3, 2) 1 e(2, n) e(n, 1) e(n, 2) e(n, 3) 1 be an n n matrix satisfying the basic hypotheses: e(j, k) > 0 for all 1 j, k n, e(j, j) = 1 for all 1 j n, e(j, l) e(j, k)e(k, l) for all 1 j, k, l n. Recall that the automorphic dilations on the group G are given by λ d (x 1,..., x n ) = (λ d1 x 1,..., λ dn x n ).
19 For 1 j n let N j (x 1,..., x n ) = x 1 e(j,1)/d1 + + x n e(j,n)/dn, Each N j is a homogeneous norm for a family of dilations on R n : N j ( λ d 1 e(j,1) x1,..., λ dn e(j,n) xn ) = λ N(x1,..., x n ). P(E) is a class of tempered distributions, depending on the matrix E and smooth away from the origin on R n, defined in terms of differential inequalities and cancellation conditions.
20 Differential inequalities: If K P(E) then away from the origin K is given by a smooth function and for each multi-index α = (α 1,..., α n ) where Note that α K(x 1,..., x n ) C α n j=1 N j (x 1,..., x n ) dj(1+αj) N j (x 1,..., x n ) = x 1 e(j,1)/d1 + + x n e(j,n)/dn. N j ( λ d 1 e(j,1) x1,..., λ dn e(j,n) xn ) = λ N(x1,..., x n ) so N j is a homogeneous norm for the family of dilations on R n given by λ j (x 1,..., x n ) = ( λ d 1 ) e(j,1) x1,..., λ dn e(j,n) xn. Derivatives of K with respect to the variable x j are controlled by the norm N j.
21 Cancellation Conditions: These are defined in terms of the action of the distribution K on dilates of normalized bump functions. Let 0 m n 1 and let ψ be any normalized bump function of n m variables. There are two requirements: If R = (R m+1,..., R n ) and each R j > 0 set ψ R (x m+1,..., x n ) = ψ(r m+1 x m+1,..., R n x n ). Define a tempered distribution K # R on Rm by setting K # R, ϕ = K, ϕ ψ R for all ϕ S(R m ). Away from the origin K # R is given by a smooth function and α 1 x 1 x αm m K # R (x 1,..., x m ) C α m j=1 with C α independent of ψ and R. N j (x 1,..., x m, 0,..., 0) aj(1+αj) The same holds for any permutation of the variables x 1,..., x n.
22 The rank of E Let E be an n n satisfying the basic hypotheses. Lemma (a) Rank (E) = 1 if and only if there is a dilation structure on R n for which P(E) is the space of Calderón-Zygmund kernels. (b) If rank (E) > 1 and K P(E) then K is integrable at infinity. (c) Suppose the rank of E is m. If K P(E) then for λ 1 { x R n : K(x) > λ} λ 1 log(λ) m 1. Moreover there exists K P(E) so that { x R n : K(x) > λ} λ 1 log(λ) m 1.
23 Characterization via the Fourier transform For rank(e) > 1 only the local behavior is important. Set { } P 0 (E) = K P(E) : K is rapidly decreasing outside the unit ball. Such distributions can be characterized by the behavior of their Fourier transform K. Recall that N j (x) = x 1 e(j,1)/d1 + + x n e(j,n)/dn, 1 j n. The dual norms are then given by Put N j (ξ) = ξ 1 1/e(1,j)d1 + + ξ n 1/e(n,j)dn, 1 j n. M (E) = {m C (R n ) : α m(ξ) n Cα Theorem K P 0 (E) if and only if K = m M (E). j=1 ( 1 + Nj (ξ) ) α jd j }.
24 Marked Partitions The analysis of distributions K P 0 (E) relies on a partition of the unit ball B(1) R n into regions where one summand is dominant in each norm N j (x). Similarly the analysis of m M (E) depends on a partition of R n \ B(1) into regions where one term in each dual norm N j (ξ) is dominant. For example, on the unit ball, the principal region S 0 is the set where for 1 j n the term x j e(j,j)/dj = x j 1/dj is dominant in N j (x) = n k=1 x k e(j,k)/d k : S 0 = { x B(1) : x k e(j,k)/d k x j 1/dj for all j, k }. Note that if x S 0 then N j (x) x j 1/dj. On this region, the differential inequalities for K P 0 (E) simplify: α K(x 1,..., x n ) n N j (x 1,..., x n ) dj(1+αj) j=1 n x j (1+αj). On the subset S 0 the differential inequalities for K are exactly the same as the differential inequalities for a product kernel. j=1
25 The following simple observation is a key to the study of other regions. Lemma Suppose x B(1) and suppose the term x k e(j,k)/d k is dominant in the norm N j (x). Then x k e(k,k)/d k = x k 1/d k is dominant in N k (x). Proof. Since x k e(j,k)/d k is dominant in the norm N j (x), x k e(j,k)/d k x l e(l,k)/d l, 1 l n. According to the basic hypothesis, e(l, k) e(l, j)e(j, k) for any 1 j, k, l n. Then since x l 1, x k 1/d k x l e(l,k)/e(j,k)d l x l e(l,j)e(j,k)/e(j,k)d l = x l e(l,j)/d l.
26 A marked partition S of {1,..., n} is a collection of disjoint subsets I 1,..., I s {1,..., n} whose union is all of {1,..., n}, together with a marked element k r I r, 1 r s. Write S = ( (I 1, k 1 ),..., (I s, k s ) ). The decomposition of B(1) is parameterized by marked partitions. For x B(1) outside a set of measure zero, there is exactly one dominant term in each norm N j (x). Let k 1,..., k s be the distinct integers which arise as subscripts of these dominant terms. Let { } I r = j {1,..., n} : x kr e(j,kr)/d kr is dominant in Nj (x), 1 r s. The subsets I 1,..., I s are disjoint, I 1 I s = {1,..., n}, and by the Lemma, k r I r. Thus S = ( (I 1, k 1 ),..., (I s, k s ) ) is a marked partition.
27 Characterization of P(E) via dyadic decompositions Let ϕ C0 (R n ). ϕ has strong cancellation if Set 1 k n. R ϕ(x 1,..., x k,..., x n ) dx k = 0 for [ϕ] I (x) = 2 n k=1 i k ϕ(2 i1 x 1,..., 2 in x n ) if I = (i 1,..., i n ) Z n. Γ Z (E) = { } (i 1,..., i n ) Z n : e(j, k)i k i j < 0 for all 1 j, k n. Lemma Let { ϕ I : I Γ(E) } C0 (R n ) be a family of normalized bump functions. Assume that each ϕ I has strong cancellation. Then K(x) = [ϕ I ] I (x) I Γ(E) converges in the sense of distributions to an element of P 0 (E).
28 The formulation of the converse assertion uses the decomposition of R n based on marked partitions. Let S = ( (I 1, k 1 ),..., (I s, k s ) ) be a marked partition. Write R n = R C1 R Cs and (x 1,..., x n ) = ( x 1,..., x s ) where R Cr is the set of variables x j R n with j I r. Then There is an s s matrix E S satisfying the basic hypotheses. There is a space of distributions P(E S ) P(E). There is a cone Γ Z (E S ) = { } (i 1,..., i n ) Z n : e S (j, k)i k i j < 0. If K P(E) then K = ψ 0 + S K S where ψ 0 S(R n ) and K S P(E S ). Moreover K S (x) = [ϕ I ] I (x). I Γ(E S )
29 The Basic Hypotheses If the principal region is not empty, then there exists (x 1,..., x n ) with each x j < 1 so that x j 1 d j xk e(j,k) d k ( x l e(k,l) ) d e(j,k) l = xl d(j,k)e(k,l) d l. (1) But since x S 0, x j x l e(j,l). (2) Unless the inequalities defining S 0 are self-improving, inequality (2) should imply (1), and so Since x l 1 it follows that x l e(j,l) x l e(j,k)e(k,l). e(j, l) e(j, k)e(k, l).
30 The basic hypotheses also arise as follows. Let F = { f(j, k) } be any n n matrix with positive entries, and let Γ(F) = { (t 1,..., t n ) R n : f(j, k)t k t j < 0 for all 1 j, k n }. Lemma If Γ(F) there exists a unique n n matrix E = { e(j, k)} with positive entries such that Γ(E) = Γ(F) and such that the entries of E satisfy Moreover, e(j, k) f(j, k). e(j, j) = 1 1 j n, e(j, l) e(j, k)e(k, l) 1 j, k, l n.
31 Two-flag kernels Consider a distribution K which is a flag kernel relative to two opposite flags with dilations λ a (x) = (λ a1 x 1,..., λ an x n ), λ b (x) = (λ b1 x 1,..., λ bn x n ). Then K satisfies appropriate cancellation conditions and the following differential inequalities: α K(x) ( ) 1 αj n j=1 x 1 aj/a1 + x 2 aj/a2 + + x j 1 aj/aj 1 + x j ) 1 αj n j=1 ( x j + x j+1 bj/bj x n 1 bj/bn 1 + x n bj/bn Note that the differential inequalities give no information if x 1 = x n = 0 and n 3.
32 Theorem Suppose that a1 b 1 a2 b 2 an b n with at least one strict inequality. (a) The function K is integrable at infinity, and we can write K = K 0 + K where K L 1 (R N ) C (R N ), and K 0 is a two-flag kernel supported in B(1). (b) The kernel K 0 belongs to the class P 0 (E) associated to the matrix 1 b 1 /b 2 b 1 /b 3 b 1 /b n 1 b 1 /b n a 2 /a 1 1 b 2 /b 3 b 2 /b n 1 b 2 /b n a 3 /a 1 a 3 /a 2 1 b 3 /b n 1 b 3 /b n E = a n 1 /a 1 a n 1 /a 2 a n 1 /a 3 1 b n 1 /b n a n /a 1 a n /a 2 a n /a 3 a n /a n 1 1
33 Convolution on groups When studying convolution on general homogeneous nilpotent Lie groups, we need to impose an additional condition on the matrix E = { e(j, k) } called double monotonicity: each row is weakly increasing from left to right and each column is weakly decreasing from top to bottom. Explicitly e(j, k) e(j, k + 1) and e(j, k) e(j + 1, k). Lemma Suppose E is doubly monotone, and let I = (i 1,..., i n ) Γ Z (E), J = (j 1,..., j n ) Γ Z (E). If ϕ, ψ are normalized bump functions, then [ϕ] I [ψ] J = [θ] K where (a) θ is a normalized bump function; (b) K = (k 1,..., k n ) Γ(E) and k m = max{i m, j m }.
34 Convolution operators are bounded on L p Theorem Let G = R n be a homogeneous nilpotent Lie group and let E be a doubly monotone matrix. If K P 0 (E) then the operator T K ϕ = ϕ K, defined initially on the Schwartz space S(R n ), extends uniquely to a bounded operator on L p (G) for 1 < p <. In fact, every kernel K P(E) is a flag kernel on an appropriate flag.
35 P(E) is an algebra under convolution Theorem Let G = R n be a homogeneous nilpotent Lie group and let E be a doubly monotone matrix. If K, L P 0 (E) then there exists M P 0 (E) such that T K T L = T M. The proof is quite technical and uses the dyadic decomposition of kernels. If S and T are marked partitions, one studies [ϕ I ] I [ψ J ] J. I Γ(E S ) J Γ(E T
36 Composition of Calderón-Zygmund kernels Let K a, K b be Calderón-Zygmund kernels associated with the dilations λ a (x) = ( λ a1 x 1,..., λ an x n ) λ b (x) = ( λ b1 x 1,..., λ bn x n ). Assume that a k a k+1 and b k b k+1. Theorem Let E = {e(j, k)} where { aj e(j, k) = max, b } j. a k b k The operator T K1 T K2 is given by convolution with a tempered distribution L with L P(E). Final Remarks: 1. For appropriate a and b the rank of E can be as large as n. 2. However L = n 1 j=1 L j where L j P(E j ) and E j has rank 2.
On a class of pseudodifferential operators with mixed homogeneities
On a class of pseudodifferential operators with mixed homogeneities Po-Lam Yung University of Oxford July 25, 2014 Introduction Joint work with E. Stein (and an outgrowth of work of Nagel-Ricci-Stein-Wainger,
More informationA new class of pseudodifferential operators with mixed homogenities
A new class of pseudodifferential operators with mixed homogenities Po-Lam Yung University of Oxford Jan 20, 2014 Introduction Given a smooth distribution of hyperplanes on R N (or more generally on a
More informationChapter One. The Calderón-Zygmund Theory I: Ellipticity
Chapter One The Calderón-Zygmund Theory I: Ellipticity Our story begins with a classical situation: convolution with homogeneous, Calderón- Zygmund ( kernels on R n. Let S n 1 R n denote the unit sphere
More informationSingular Integrals. 1 Calderon-Zygmund decomposition
Singular Integrals Analysis III Calderon-Zygmund decomposition Let f be an integrable function f dx 0, f = g + b with g Cα almost everywhere, with b
More informationSCHWARTZ SPACES ASSOCIATED WITH SOME NON-DIFFERENTIAL CONVOLUTION OPERATORS ON HOMOGENEOUS GROUPS
C O L L O Q U I U M M A T H E M A T I C U M VOL. LXIII 1992 FASC. 2 SCHWARTZ SPACES ASSOCIATED WITH SOME NON-DIFFERENTIAL CONVOLUTION OPERATORS ON HOMOGENEOUS GROUPS BY JACEK D Z I U B A Ń S K I (WROC
More informationWe denote the space of distributions on Ω by D ( Ω) 2.
Sep. 1 0, 008 Distributions Distributions are generalized functions. Some familiarity with the theory of distributions helps understanding of various function spaces which play important roles in the study
More informationMicrolocal Methods in X-ray Tomography
Microlocal Methods in X-ray Tomography Plamen Stefanov Purdue University Lecture I: Euclidean X-ray tomography Mini Course, Fields Institute, 2012 Plamen Stefanov (Purdue University ) Microlocal Methods
More informationClassical Fourier Analysis
Loukas Grafakos Classical Fourier Analysis Second Edition 4y Springer 1 IP Spaces and Interpolation 1 1.1 V and Weak IP 1 1.1.1 The Distribution Function 2 1.1.2 Convergence in Measure 5 1.1.3 A First
More informationESTIMATES FOR MAXIMAL SINGULAR INTEGRALS
ESTIMATES FOR MAXIMAL SINGULAR INTEGRALS LOUKAS GRAFAKOS Abstract. It is shown that maximal truncations of nonconvolution L -bounded singular integral operators with kernels satisfying Hörmander s condition
More informationRESULTS ON FOURIER MULTIPLIERS
RESULTS ON FOURIER MULTIPLIERS ERIC THOMA Abstract. The problem of giving necessary and sufficient conditions for Fourier multipliers to be bounded on L p spaces does not have a satisfactory answer for
More informationMath The Laplacian. 1 Green s Identities, Fundamental Solution
Math. 209 The Laplacian Green s Identities, Fundamental Solution Let be a bounded open set in R n, n 2, with smooth boundary. The fact that the boundary is smooth means that at each point x the external
More informationReal Analysis Prelim Questions Day 1 August 27, 2013
Real Analysis Prelim Questions Day 1 August 27, 2013 are 5 questions. TIME LIMIT: 3 hours Instructions: Measure and measurable refer to Lebesgue measure µ n on R n, and M(R n ) is the collection of measurable
More informationMORE NOTES FOR MATH 823, FALL 2007
MORE NOTES FOR MATH 83, FALL 007 Prop 1.1 Prop 1. Lemma 1.3 1. The Siegel upper half space 1.1. The Siegel upper half space and its Bergman kernel. The Siegel upper half space is the domain { U n+1 z C
More informationHARMONIC ANALYSIS. Date:
HARMONIC ANALYSIS Contents. Introduction 2. Hardy-Littlewood maximal function 3. Approximation by convolution 4. Muckenhaupt weights 4.. Calderón-Zygmund decomposition 5. Fourier transform 6. BMO (bounded
More informationAnalytic families of multilinear operators
Analytic families of multilinear operators Mieczysław Mastyło Adam Mickiewicz University in Poznań Nonlinar Functional Analysis Valencia 17-20 October 2017 Based on a joint work with Loukas Grafakos M.
More informationHarmonic Analysis Homework 5
Harmonic Analysis Homework 5 Bruno Poggi Department of Mathematics, University of Minnesota November 4, 6 Notation Throughout, B, r is the ball of radius r with center in the understood metric space usually
More informationA NOTE ON THE HEAT KERNEL ON THE HEISENBERG GROUP
A NOTE ON THE HEAT KERNEL ON THE HEISENBERG GROUP ADAM SIKORA AND JACEK ZIENKIEWICZ Abstract. We describe the analytic continuation of the heat ernel on the Heisenberg group H n (R. As a consequence, we
More informationThe oblique derivative problem for general elliptic systems in Lipschitz domains
M. MITREA The oblique derivative problem for general elliptic systems in Lipschitz domains Let M be a smooth, oriented, connected, compact, boundaryless manifold of real dimension m, and let T M and T
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More informationLaplace s Equation. Chapter Mean Value Formulas
Chapter 1 Laplace s Equation Let be an open set in R n. A function u C 2 () is called harmonic in if it satisfies Laplace s equation n (1.1) u := D ii u = 0 in. i=1 A function u C 2 () is called subharmonic
More information8 Singular Integral Operators and L p -Regularity Theory
8 Singular Integral Operators and L p -Regularity Theory 8. Motivation See hand-written notes! 8.2 Mikhlin Multiplier Theorem Recall that the Fourier transformation F and the inverse Fourier transformation
More informationSingular integrals with flag kernels on homogeneous groups, I
Rev. Mat. Iberoam. 28 (2012), no. 3, 631 722 doi 10.4171/rmi/688 c European Mathematical Society Singular integrals with flag kernels on homogeneous groups, I Alexander Nagel, Fulvio Ricci, Elias M. Stein
More informationA BRIEF INTRODUCTION TO SEVERAL COMPLEX VARIABLES
A BRIEF INTRODUCTION TO SEVERAL COMPLEX VARIABLES PO-LAM YUNG Contents 1. The Cauchy-Riemann complex 1 2. Geometry of the domain: Pseudoconvexity 3 3. Solvability of the Cauchy-Riemann operator 5 4. The
More informationAverage theorem, Restriction theorem and Strichartz estimates
Average theorem, Restriction theorem and trichartz estimates 2 August 27 Abstract We provide the details of the proof of the average theorem and the restriction theorem. Emphasis has been placed on the
More informationThe Hilbert transform
The Hilbert transform Definition and properties ecall the distribution pv(, defined by pv(/(ϕ := lim ɛ ɛ ϕ( d. The Hilbert transform is defined via the convolution with pv(/, namely (Hf( := π lim f( t
More informationOverview of normed linear spaces
20 Chapter 2 Overview of normed linear spaces Starting from this chapter, we begin examining linear spaces with at least one extra structure (topology or geometry). We assume linearity; this is a natural
More informationA SHORT PROOF OF THE COIFMAN-MEYER MULTILINEAR THEOREM
A SHORT PROOF OF THE COIFMAN-MEYER MULTILINEAR THEOREM CAMIL MUSCALU, JILL PIPHER, TERENCE TAO, AND CHRISTOPH THIELE Abstract. We give a short proof of the well known Coifman-Meyer theorem on multilinear
More informationA review: The Laplacian and the d Alembertian. j=1
Chapter One A review: The Laplacian and the d Alembertian 1.1 THE LAPLACIAN One of the main goals of this course is to understand well the solution of wave equation both in Euclidean space and on manifolds
More informationJUHA KINNUNEN. Harmonic Analysis
JUHA KINNUNEN Harmonic Analysis Department of Mathematics and Systems Analysis, Aalto University 27 Contents Calderón-Zygmund decomposition. Dyadic subcubes of a cube.........................2 Dyadic cubes
More informationRegularizations of Singular Integral Operators (joint work with C. Liaw)
1 Outline Regularizations of Singular Integral Operators (joint work with C. Liaw) Sergei Treil Department of Mathematics Brown University April 4, 2014 2 Outline 1 Examples of Calderón Zygmund operators
More informationHandlebody Decomposition of a Manifold
Handlebody Decomposition of a Manifold Mahuya Datta Statistics and Mathematics Unit Indian Statistical Institute, Kolkata mahuya@isical.ac.in January 12, 2012 contents Introduction What is a handlebody
More information1 Continuity Classes C m (Ω)
0.1 Norms 0.1 Norms A norm on a linear space X is a function : X R with the properties: Positive Definite: x 0 x X (nonnegative) x = 0 x = 0 (strictly positive) λx = λ x x X, λ C(homogeneous) x + y x +
More informationDecouplings and applications
April 27, 2018 Let Ξ be a collection of frequency points ξ on some curved, compact manifold S of diameter 1 in R n (e.g. the unit sphere S n 1 ) Let B R = B(c, R) be a ball with radius R 1. Let also a
More informationRIEMANN SURFACES. max(0, deg x f)x.
RIEMANN SURFACES 10. Weeks 11 12: Riemann-Roch theorem and applications 10.1. Divisors. The notion of a divisor looks very simple. Let X be a compact Riemann surface. A divisor is an expression a x x x
More informationYour first day at work MATH 806 (Fall 2015)
Your first day at work MATH 806 (Fall 2015) 1. Let X be a set (with no particular algebraic structure). A function d : X X R is called a metric on X (and then X is called a metric space) when d satisfies
More informationRecall that if X is a compact metric space, C(X), the space of continuous (real-valued) functions on X, is a Banach space with the norm
Chapter 13 Radon Measures Recall that if X is a compact metric space, C(X), the space of continuous (real-valued) functions on X, is a Banach space with the norm (13.1) f = sup x X f(x). We want to identify
More information1 Directional Derivatives and Differentiability
Wednesday, January 18, 2012 1 Directional Derivatives and Differentiability Let E R N, let f : E R and let x 0 E. Given a direction v R N, let L be the line through x 0 in the direction v, that is, L :=
More informationContents Introduction and Review Boundary Behavior The Heisenberg Group Analysis on the Heisenberg Group
Contents 1 Introduction and Review... 1 1.1 Harmonic Analysis on the Disc... 1 1.1.1 The Boundary Behavior of Holomorphic Functions... 4 Exercises... 15 2 Boundary Behavior... 19 2.1 The Modern Era...
More informationPseudo-Poincaré Inequalities and Applications to Sobolev Inequalities
Pseudo-Poincaré Inequalities and Applications to Sobolev Inequalities Laurent Saloff-Coste Abstract Most smoothing procedures are via averaging. Pseudo-Poincaré inequalities give a basic L p -norm control
More informationThe wave front set of a distribution
The wave front set of a distribution The Fourier transform of a smooth compactly supported function u(x) decays faster than any negative power of the dual variable ξ; that is for every number N there exists
More informationYour first day at work MATH 806 (Fall 2015)
Your first day at work MATH 806 (Fall 2015) 1. Let X be a set (with no particular algebraic structure). A function d : X X R is called a metric on X (and then X is called a metric space) when d satisfies
More informationScattered Data Interpolation with Polynomial Precision and Conditionally Positive Definite Functions
Chapter 3 Scattered Data Interpolation with Polynomial Precision and Conditionally Positive Definite Functions 3.1 Scattered Data Interpolation with Polynomial Precision Sometimes the assumption on the
More informationCauchy Integrals and Cauchy-Szegö Projections. Elias M. Stein. Conference in honor of Michael Christ Madison, May 2016
1 Cauchy Integrals and Cauchy-Szegö Projections Elias M. Stein Conference in honor of Michael Christ Madison, May 2016 The goal: In C n, to study the L p properties of Cauchy integrals and the Cauchy-Szegö
More informationQUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday September 21, 2004 (Day 1)
QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday September 21, 2004 (Day 1) Each of the six questions is worth 10 points. 1) Let H be a (real or complex) Hilbert space. We say
More informationA Tilt at TILFs. Rod Nillsen University of Wollongong. This talk is dedicated to Gary H. Meisters
A Tilt at TILFs Rod Nillsen University of Wollongong This talk is dedicated to Gary H. Meisters Abstract In this talk I will endeavour to give an overview of some aspects of the theory of Translation Invariant
More informationL p -boundedness of the Hilbert transform
L p -boundedness of the Hilbert transform Kunal Narayan Chaudhury Abstract The Hilbert transform is essentially the only singular operator in one dimension. This undoubtedly makes it one of the the most
More informationLocally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem
56 Chapter 7 Locally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem Recall that C(X) is not a normed linear space when X is not compact. On the other hand we could use semi
More informationTopics in Harmonic Analysis Lecture 6: Pseudodifferential calculus and almost orthogonality
Topics in Harmonic Analysis Lecture 6: Pseudodifferential calculus and almost orthogonality Po-Lam Yung The Chinese University of Hong Kong Introduction While multiplier operators are very useful in studying
More informationDISCRETE LITTLEWOOD-PALEY-STEIN THEORY AND MULTI-PARAMETER HARDY SPACES ASSOCIATED WITH FLAG SINGULAR INTEGRALS
DSCRETE LTTLEWOOD-PALEY-STEN THEORY AND MULT-PARAMETER HARDY SPACES ASSOCATED WTH LAG SNGULAR NTEGRALS Yongsheng Han Department of Mathematics Auburn University Auburn, AL 36849, U.S.A E-mail: hanyong@mail.auburn.edu
More informationSingular integrals on NA groups
Singular integrals on NA groups joint work with Alessio Martini and Alessandro Ottazzi Maria Vallarino Dipartimento di Scienze Matematiche Politecnico di Torino LMS Midlands Regional Meeting and Workshop
More informationExact fundamental solutions
Journées Équations aux dérivées partielles Saint-Jean-de-Monts, -5 juin 998 GDR 5 (CNRS) Exact fundamental solutions Richard Beals Abstract Exact fundamental solutions are known for operators of various
More informationM ath. Res. Lett. 17 (2010), no. 04, c International Press 2010 A NOTE ON TWISTED DISCRETE SINGULAR RADON. Lillian B.
M ath. Res. Lett. 17 (2010), no. 04, 701 720 c International Press 2010 A NOTE ON TWISTED DISCRETE SINGULAR RADON TRANSFORMS Lillian B. Pierce Abstract. In this paper we consider three types of discrete
More information4 Riesz Kernels. Since the functions k i (ξ) = ξ i. are bounded functions it is clear that R
4 Riesz Kernels. A natural generalization of the Hilbert transform to higher dimension is mutiplication of the Fourier Transform by homogeneous functions of degree 0, the simplest ones being R i f(ξ) =
More informationMultilinear local Tb Theorem for Square functions
Multilinear local Tb Theorem for Square functions (joint work with J. Hart and L. Oliveira) September 19, 2013. Sevilla. Goal Tf(x) L p (R n ) C f L p (R n ) Goal Tf(x) L p (R n ) C f L p (R n ) Examples
More informationSome examples of quasiisometries of nilpotent Lie groups
Some examples of quasiisometries of nilpotent Lie groups Xiangdong Xie Abstract We construct quasiisometries of nilpotent Lie groups. In particular, for any simply connected nilpotent Lie group N, we construct
More informationRESTRICTION. Alex Iosevich. Section 0: Introduction.. A natural question to ask is, does the boundedness of R : L 2(r+1)
FOURIER TRANSFORM, L 2 THEOREM, AND SCALING RESTRICTION Alex Iosevich Abstract. We show, using a Knapp-type homogeneity argument, that the (L p, L 2 ) restriction theorem implies a growth condition on
More informationDuality of multiparameter Hardy spaces H p on spaces of homogeneous type
Duality of multiparameter Hardy spaces H p on spaces of homogeneous type Yongsheng Han, Ji Li, and Guozhen Lu Department of Mathematics Vanderbilt University Nashville, TN Internet Analysis Seminar 2012
More informationOn some weighted fractional porous media equations
On some weighted fractional porous media equations Gabriele Grillo Politecnico di Milano September 16 th, 2015 Anacapri Joint works with M. Muratori and F. Punzo Gabriele Grillo Weighted Fractional PME
More informationErrata Applied Analysis
Errata Applied Analysis p. 9: line 2 from the bottom: 2 instead of 2. p. 10: Last sentence should read: The lim sup of a sequence whose terms are bounded from above is finite or, and the lim inf of a sequence
More informationFourier Transform & Sobolev Spaces
Fourier Transform & Sobolev Spaces Michael Reiter, Arthur Schuster Summer Term 2008 Abstract We introduce the concept of weak derivative that allows us to define new interesting Hilbert spaces the Sobolev
More informationOptimization Theory. A Concise Introduction. Jiongmin Yong
October 11, 017 16:5 ws-book9x6 Book Title Optimization Theory 017-08-Lecture Notes page 1 1 Optimization Theory A Concise Introduction Jiongmin Yong Optimization Theory 017-08-Lecture Notes page Optimization
More informationNew Proof of Hörmander multiplier Theorem on compact manifolds without boundary
New Proof of Hörmander multiplier Theorem on compact manifolds without boundary Xiangjin Xu Department of athematics Johns Hopkins University Baltimore, D, 21218, USA xxu@math.jhu.edu Abstract On compact
More informationDouble Layer Potentials on Polygons and Pseudodifferential Operators on Lie Groupoids
Double Layer Potentials on Polygons and Pseudodifferential Operators on Lie Groupoids joint work with Hengguang Li Yu Qiao School of Mathematics and Information Science Shaanxi Normal University Xi an,
More informationMAXIMUM REGULARITY OF HUA HARMONIC FUNCTIONS
MAXIMUM REGULARITY OF UA ARMONIC FUNCTIONS P. JAMING 1. The Siegel upper half space 1.1. Setting. On C 2 consider r(z) = Im(z 2 ) z 1 2. The Siegel upper half space and its boundary are given by U ={z
More informationRemarks on Extremization Problems Related To Young s Inequality
Remarks on Extremization Problems Related To Young s Inequality Michael Christ University of California, Berkeley University of Wisconsin May 18, 2016 Part 1: Introduction Young s convolution inequality
More informationReal Analysis Problems
Real Analysis Problems Cristian E. Gutiérrez September 14, 29 1 1 CONTINUITY 1 Continuity Problem 1.1 Let r n be the sequence of rational numbers and Prove that f(x) = 1. f is continuous on the irrationals.
More informationWeighted norm inequalities for singular integral operators
Weighted norm inequalities for singular integral operators C. Pérez Journal of the London mathematical society 49 (994), 296 308. Departmento de Matemáticas Universidad Autónoma de Madrid 28049 Madrid,
More informationClassification of root systems
Classification of root systems September 8, 2017 1 Introduction These notes are an approximate outline of some of the material to be covered on Thursday, April 9; Tuesday, April 14; and Thursday, April
More informationQualifying Exams I, Jan where µ is the Lebesgue measure on [0,1]. In this problems, all functions are assumed to be in L 1 [0,1].
Qualifying Exams I, Jan. 213 1. (Real Analysis) Suppose f j,j = 1,2,... and f are real functions on [,1]. Define f j f in measure if and only if for any ε > we have lim µ{x [,1] : f j(x) f(x) > ε} = j
More informationGeometry and the Kato square root problem
Geometry and the Kato square root problem Lashi Bandara Centre for Mathematics and its Applications Australian National University 29 July 2014 Geometric Analysis Seminar Beijing International Center for
More informationTHE MULTIPLICATIVE ERGODIC THEOREM OF OSELEDETS
THE MULTIPLICATIVE ERGODIC THEOREM OF OSELEDETS. STATEMENT Let (X, µ, A) be a probability space, and let T : X X be an ergodic measure-preserving transformation. Given a measurable map A : X GL(d, R),
More informationMELIN CALCULUS FOR GENERAL HOMOGENEOUS GROUPS
Arkiv för matematik, 45 2007, 31-48 Revised March 16, 2013 MELIN CALCULUS FOR GENERAL HOMOGENEOUS GROUPS PAWE L G LOWACKI WROC LAW Abstract. The purpose of this note is to give an extension of the symbolic
More informationLecture 2: Some basic principles of the b-calculus
Lecture 2: Some basic principles of the b-calculus Daniel Grieser (Carl von Ossietzky Universität Oldenburg) September 20, 2012 Summer School Singular Analysis Daniel Grieser (Oldenburg) Lecture 2: Some
More informationSTRONGLY SINGULAR INTEGRALS ALONG CURVES. 1. Introduction. It is standard and well known that the Hilbert transform along curves: f(x γ(t)) dt
STRONGLY SINGULAR INTEGRALS ALONG CURVES NORBERTO LAGHI NEIL LYALL Abstract. In this article we obtain L 2 bounds for strongly singular integrals along curves in R d ; our results both generalise and extend
More informationChanging sign solutions for the CR-Yamabe equation
Changing sign solutions for the CR-Yamabe equation Ali Maalaoui (1) & Vittorio Martino (2) Abstract In this paper we prove that the CR-Yamabe equation on the Heisenberg group has infinitely many changing
More informationA RELATIONSHIP BETWEEN THE DIRICHLET AND REGULARITY PROBLEMS FOR ELLIPTIC EQUATIONS. Zhongwei Shen
A RELATIONSHIP BETWEEN THE DIRICHLET AND REGULARITY PROBLEMS FOR ELLIPTIC EQUATIONS Zhongwei Shen Abstract. Let L = diva be a real, symmetric second order elliptic operator with bounded measurable coefficients.
More informationMATH 825 FALL 2014 ANALYSIS AND GEOMETRY IN CARNOT-CARATHÉODORY SPACES
MATH 825 FALL 2014 ANALYSIS AND GEOMETRY IN CARNOT-CARATHÉODORY SPACES Contents 1. Introduction 1 1.1. Vector fields and flows 2 1.2. Metrics 2 1.3. Commutators 3 1.4. Consequences of Hörmander s condition
More informationHere we used the multiindex notation:
Mathematics Department Stanford University Math 51H Distributions Distributions first arose in solving partial differential equations by duality arguments; a later related benefit was that one could always
More informationProblem: A class of dynamical systems characterized by a fast divergence of the orbits. A paradigmatic example: the Arnold cat.
À È Ê ÇÄÁ Ë ËÌ ÅË Problem: A class of dynamical systems characterized by a fast divergence of the orbits A paradigmatic example: the Arnold cat. The closure of a homoclinic orbit. The shadowing lemma.
More informationA NEW PROOF OF THE ATOMIC DECOMPOSITION OF HARDY SPACES
A NEW PROOF OF THE ATOMIC DECOMPOSITION OF HARDY SPACES S. DEKEL, G. KERKYACHARIAN, G. KYRIAZIS, AND P. PETRUSHEV Abstract. A new proof is given of the atomic decomposition of Hardy spaces H p, 0 < p 1,
More informationBoolean Inner-Product Spaces and Boolean Matrices
Boolean Inner-Product Spaces and Boolean Matrices Stan Gudder Department of Mathematics, University of Denver, Denver CO 80208 Frédéric Latrémolière Department of Mathematics, University of Denver, Denver
More informationChapter 2 Convex Analysis
Chapter 2 Convex Analysis The theory of nonsmooth analysis is based on convex analysis. Thus, we start this chapter by giving basic concepts and results of convexity (for further readings see also [202,
More informationSingular and maximal Radon transforms: Analysis and geometry
Annals of Mathematics, 150 (1999), 489 577 Singular and maximal Radon transforms: Analysis and geometry By Michael Christ, Alexander Nagel, Elias M. Stein, and Stephen Wainger 0. Introduction Table of
More informationCones of measures. Tatiana Toro. University of Washington. Quantitative and Computational Aspects of Metric Geometry
Cones of measures Tatiana Toro University of Washington Quantitative and Computational Aspects of Metric Geometry Based on joint work with C. Kenig and D. Preiss Tatiana Toro (University of Washington)
More information6.3 Fundamental solutions in d
6.3. Fundamental solutions in d 5 6.3 Fundamental solutions in d Since Lapalce equation is invariant under translations, and rotations (see Exercise 6.4), we look for solutions to Laplace equation having
More informationIsodiametric problem in Carnot groups
Conference Geometric Measure Theory Université Paris Diderot, 12th-14th September 2012 Isodiametric inequality in R n Isodiametric inequality: where ω n = L n (B(0, 1)). L n (A) 2 n ω n (diam A) n Isodiametric
More informationImplications of the Constant Rank Constraint Qualification
Mathematical Programming manuscript No. (will be inserted by the editor) Implications of the Constant Rank Constraint Qualification Shu Lu Received: date / Accepted: date Abstract This paper investigates
More information3. (a) What is a simple function? What is an integrable function? How is f dµ defined? Define it first
Math 632/6321: Theory of Functions of a Real Variable Sample Preinary Exam Questions 1. Let (, M, µ) be a measure space. (a) Prove that if µ() < and if 1 p < q
More informationShort note on compact operators - Monday 24 th March, Sylvester Eriksson-Bique
Short note on compact operators - Monday 24 th March, 2014 Sylvester Eriksson-Bique 1 Introduction In this note I will give a short outline about the structure theory of compact operators. I restrict attention
More informationElliptic Problems for Pseudo Differential Equations in a Polyhedral Cone
Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 9, Number 2, pp. 227 237 (2014) http://campus.mst.edu/adsa Elliptic Problems for Pseudo Differential Equations in a Polyhedral Cone
More information1. Bounded linear maps. A linear map T : E F of real Banach
DIFFERENTIABLE MAPS 1. Bounded linear maps. A linear map T : E F of real Banach spaces E, F is bounded if M > 0 so that for all v E: T v M v. If v r T v C for some positive constants r, C, then T is bounded:
More informationRIESZ BASES AND UNCONDITIONAL BASES
In this paper we give a brief introduction to adjoint operators on Hilbert spaces and a characterization of the dual space of a Hilbert space. We then introduce the notion of a Riesz basis and give some
More informationON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS
Bendikov, A. and Saloff-Coste, L. Osaka J. Math. 4 (5), 677 7 ON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS ALEXANDER BENDIKOV and LAURENT SALOFF-COSTE (Received March 4, 4)
More informationAn inverse source problem in optical molecular imaging
An inverse source problem in optical molecular imaging Plamen Stefanov 1 Gunther Uhlmann 2 1 2 University of Washington Formulation Direct Problem Singular Operators Inverse Problem Proof Conclusion Figure:
More informationMATH MEASURE THEORY AND FOURIER ANALYSIS. Contents
MATH 3969 - MEASURE THEORY AND FOURIER ANALYSIS ANDREW TULLOCH Contents 1. Measure Theory 2 1.1. Properties of Measures 3 1.2. Constructing σ-algebras and measures 3 1.3. Properties of the Lebesgue measure
More informationIntegral Representation Formula, Boundary Integral Operators and Calderón projection
Integral Representation Formula, Boundary Integral Operators and Calderón projection Seminar BEM on Wave Scattering Franziska Weber ETH Zürich October 22, 2010 Outline Integral Representation Formula Newton
More informationAn introduction to some aspects of functional analysis
An introduction to some aspects of functional analysis Stephen Semmes Rice University Abstract These informal notes deal with some very basic objects in functional analysis, including norms and seminorms
More informationi=1 α i. Given an m-times continuously
1 Fundamentals 1.1 Classification and characteristics Let Ω R d, d N, d 2, be an open set and α = (α 1,, α d ) T N d 0, N 0 := N {0}, a multiindex with α := d i=1 α i. Given an m-times continuously differentiable
More informationB 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X.
Math 6342/7350: Topology and Geometry Sample Preliminary Exam Questions 1. For each of the following topological spaces X i, determine whether X i and X i X i are homeomorphic. (a) X 1 = [0, 1] (b) X 2
More informationB. Appendix B. Topological vector spaces
B.1 B. Appendix B. Topological vector spaces B.1. Fréchet spaces. In this appendix we go through the definition of Fréchet spaces and their inductive limits, such as they are used for definitions of function
More information